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Calabi-Yau Pairs and Mirror Symmetry for Fano Varieties

Fano 品种的 Calabi-Yau 对和镜像对称性

基本信息

批准号：
EP/P029949/1
负责人：
Anne-Sophie Kaloghiros
金额：
$ 11.7万
依托单位：
Brunel University London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP029949%2F1
关键词：
Calabi Yau Pairs Mirror Symmetry

项目摘要

Algebraic geometry has for many decades been one of the core disciplines of mathematics, and the subject remains as vital today as it was 150 years ago as a source of new ideas and important problems. The basic objects studied in algebraic geometry are geometric shapes defined by polynomial equations in an ambient projective space. The Minimal Model Program (MMP) shows that, up to surgery, these shapes can be constructed out of "building blocks" of pure geometric type, and these have:(1) positive curvature (Fano varieties),(2) zero curvature (Calabi-Yau varieties),(3) negative curvature (varieties of general type).These pure geometric types correspond intuitively to the geometry of the sphere, of the Euclidian plane and of the hyperbolic plane. The geometry of each pure type has distinct features and properties: presence of rational curves, behaviour in families, surgery operations between the variety and other varieties.. A very useful technique in recent developments has been to consider perturbations of these pure geometric types. The perturbed geometric types are defined for pairs of a variety and some lower dimensional shape lying on it (for example a slice of the original shape). The varieties of perturbed pure geometric types are called log Fano, log Calabi-Yau, and of log general type. They share many of the features of the associated pure types. The geometry of pairs is very rich: a pair can have a certain perturbed type (log Calabi-Yau for example) while its underlying variety has a different pure geometric type (Fano in our example). The geometry of the pair then blends features of Calabi-Yau and Fano geometries. My research concentrates on varieties and pairs whose geometry or perturbed geometry is of Fano or Calabi-Yau type. These are important in mathematical physics: according to string theory, the fundamental objects in physics are strings rather than point-like particles. These strings move in a background that, in addition to space and time, has extra hidden dimensions curled up in a background variety which is Fano or Calabi-Yau (depending on the version of the theory). Explicitly, this proposal is concerned with the geometry of log Calabi-Yau and Fano shapes. The first project studies transformations of log Calabi-Yau shapes that preserve an additional invariant. The most interesting case is that of transformations between log Calabi-Yau shapes that are made of a Fano shape and a Calabi-Yau slice of it. The transformations are then required to preserve the volume of the Calabi-Yau slice. The second project focusses explicitly on mirror symmetry for Fano varieties. Mirror symmetry is a duality that occurs when two mathematically different background geometries produce the same physics; the two geometries are then called mirror dual. Conjecturally, Fano shapes are mirror symmetric to so called cluster varieties. Cluster varieties are families of log Calabi-Yau shapes that can be glued by the transformations studied in the first project. The proposed research will apply techniques and ideas from algebraic geometry and from the Minimal Model Program to deepen our understanding of cluster varieties and of log Calabi-Yau geometries. In turn, the results of this work will provide a new angle on the geometry of Fano shapes.

几十年来，代数几何一直是数学的核心学科之一，这门学科在今天仍然像150年前一样重要，是新思想和重要问题的来源。代数几何研究的基本对象是由周围射影空间中的多项式方程定义的几何形状。最小模型程序（MMP）表明，直到外科手术，这些形状可以由纯几何类型的“构建块”构建，并且这些具有：（1）正曲率（Fano变种），（2）零曲率（Calabi-Yau变种），（3）负曲率（一般类型的变种）。这些纯几何类型直观地对应于球面、欧几里得平面和双曲平面的几何。每个纯类型的几何有不同的特点和属性：存在合理的曲线，行为的家庭，外科手术之间的品种和其他品种。在最近的发展中，一个非常有用的技术是考虑这些纯几何类型的扰动。扰动的几何类型被定义为一个品种和一些低维形状躺在它（例如一个切片的原始形状）的对。扰动纯几何型的变种称为对数Fano型、对数Calabi-Yau型和对数一般型。它们共享相关纯类型的许多特性。偶的几何是非常丰富的：一个偶可以有一个特定的扰动类型（例如对数卡-丘），而它的基础变体有一个不同的纯几何类型（在我们的例子中是法诺）。然后，这对的几何形状融合了Calabi-Yau和Fano几何形状的特征。我的研究集中在品种和对的几何或扰动几何是Fano或Calabi-Yau型。这些在数学物理学中很重要：根据弦理论，物理学中的基本对象是弦而不是点状粒子。这些弦在一个背景中运动，除了空间和时间之外，还有额外的隐藏维蜷缩在一个背景变体中，这个变体是法诺或卡拉比-丘（取决于理论的版本）。解释，这个建议是有关的几何形状的日志卡-丘和法诺形状。第一个项目研究保持额外不变的对数卡-丘形状的变换。最有趣的情况是对数卡-丘形状之间的变换，对数卡-丘形状由Fano形状和它的卡-丘切片组成，然后要求变换保持卡-丘切片的体积。第二个项目的重点是明确的镜像对称的法诺品种。镜像对称（英语：Mirror symmetry）是一种对偶，当两个数学上不同的背景几何产生相同的物理时，这两个几何就被称为镜像对偶。从理论上讲，Fano形状与所谓的簇簇变体是镜像对称的。簇的品种是家庭的日志卡-丘形状，可以胶合的第一个项目中研究的转换。拟议的研究将应用技术和思想，从代数几何和最小模型程序，以加深我们的理解集群品种和日志卡-丘几何。反过来，这项工作的结果将提供一个新的角度对Fano形状的几何形状。